SMO算法，platt论文的原始算法及优化算法

 platt论文：[PDF] Sequential Minimal Optimization : A Fast Algorithm for Training Support Vector Machines | Semantic Scholar

算法优化：[PDF] Improvements to Platt's SMO Algorithm for SVM Classifier Design | Semantic Scholar

包含个人platt版SMO代码实现、SMO 优化算法、libsvm：yanzhi0922/SVM: 2024.07.12 (github.com)

数据集获取：LIBSVM Data: Classification, Regression, and Multi-label (ntu.edu.tw)

platt原论文SMO伪代码：

target = desired output vector
point = training point matrix
procedure takeStep(i1,i2)
    if (i1 == i2) return 0
    alph1 = Lagrange multiplier for i1
    y1 = target[i1]
    E1 = SVM output on point[i1] – y1 (check in error cache)
    s = y1*y2
    Compute L, H via equations (13) and (14)
    if (L == H)
        return 0
    k11 = kernel(point[i1],point[i1])
    k12 = kernel(point[i1],point[i2])
    k22 = kernel(point[i2],point[i2])
    eta = k11+k22-2*k12
    if (eta > 0)
    {
        a2 = alph2 + y2*(E1-E2)/eta
        if (a2 < L) a2 = L
        else if (a2 > H) a2 = H
    }
    else
    {
        Lobj = objective function at a2=L
        Hobj = objective function at a2=H
        if (Lobj < Hobj-eps)
            a2 = L
        else if (Lobj > Hobj+eps)
            a2 = H
        else
            a2 = alph2
    }
    if (|a2-alph2| < eps*(a2+alph2+eps))
        return 0
    a1 = alph1+s*(alph2-a2)
    Update threshold to reflect change in Lagrange multipliers
    Update weight vector to reflect change in a1 & a2, if SVM is linear
    Update error cache using new Lagrange multipliers
    Store a1 in the alpha array
    Store a2 in the alpha array
    return 1
endprocedure

procedure examineExample(i2)
    y2 = target[i2]
    alph2 = Lagrange multiplier for i2
    E2 = SVM output on point[i2] – y2 (check in error cache)
    r2 = E2*y2
    if ((r2 < -tol && alph2 < C) || (r2 > tol && alph2 > 0))
    {
        if (number of non-zero & non-C alpha > 1)
        {
            i1 = result of second choice heuristic (section 2.2)
            if takeStep(i1,i2)
                return 1
        }
        loop over all non-zero and non-C alpha, starting at a random point
        {
            i1 = identity of current alpha
            if takeStep(i1,i2)
                return 1
        }
        loop over all possible i1, starting at a random point
        {
            i1 = loop variable
            if (takeStep(i1,i2)
                return 1
        }
    }
    return 0
endprocedure

main routine:
    numChanged = 0;
    examineAll = 1;
    while (numChanged > 0 | examineAll)
    {
        numChanged = 0;
        if (examineAll)
            loop I over all training examples
                numChanged += examineExample(I)
        else
            loop I over examples where alpha is not 0 & not C
                numChanged += examineExample(I)
        if (examineAll == 1)
            examineAll = 0
        else if (numChanged == 0)
            examineAll = 1
}

以上算法基本原理相同，结果相同，优化是时间复杂度上的优化 ，libsvm时间复杂度最优